# PrimePi(10^23): A006880 and MathWorld disagreement

Maximilian Hasler maximilian.hasler at gmail.com
Wed Feb 27 01:46:40 CET 2008

```As promised, the complete table including R(x)-pi(x) in the last column.
For the records, I include the few lines of PARI code needed to reproduce it.
Regards,
Maximilian

a6880 = [4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534,
455052511, 4118054813, 37607912018, 346065536839, 3204941750802,
29844570422669, 279238341033925, 2623557157654233, 24739954287740860,
234057667276344607, 2220819602560918840, 21127269486018731928,
201467286689315906290, 1925320391606803968923];
R(x) = 1+suminf(k=1, log(x)^k/(k*k!*zeta(k+1)))
vector(#a6880,i,[Str("10^{",i,"}"),a6880[i],round(10^i/log(10^i)-a6880[i]),round(-eint1(-log(10^i))-a6880[i]),round(R(10^i)-a6880[i])]);
printtex(Mat(%~))
\$\$ \matrix{
10^{1}&4&0&2&1\cr
10^{2}&25&-3&5&1\cr
10^{3}&168&-23&10&0\cr
10^{4}&1229&-143&17&-2\cr
10^{5}&9592&-906&38&-5\cr
10^{6}&78498&-6116&130&29\cr
10^{7}&664579&-44158&339&88\cr
10^{8}&5761455&-332774&754&97\cr
10^{9}&50847534&-2592592&1701&-79\cr
10^{10}&455052511&-20758029&3104&-1828\cr
10^{11}&4118054813&-169923159&11588&-2318\cr
10^{12}&37607912018&-1416705193&38263&-1476\cr
10^{13}&346065536839&-11992858452&108971&-5773\cr
10^{14}&3204941750802&-102838308636&314890&-19200\cr
10^{15}&29844570422669&-891604962452&1052619&73218\cr
10^{16}&279238341033925&-7804289844393&3214632&327052\cr
10^{17}&2623557157654233&-68883734693928&7956589&-598255\cr
10^{18}&24739954287740860&-612483070893536&21949555&-3501366\cr
10^{19}&234057667276344607&-5481624169369961&99877775&23884333\cr
10^{20}&2220819602560918840&-49347193044659702&222744644&-4891825\cr
10^{21}&21127269486018731928&-446579871578168707&597394254&-86432204\cr
10^{22}&201467286689315906290&-4060704006019620994&1932355208&-127132665\cr
10^{23}&1925320391606803968923&-37083513766578631309&7250186216&1033299853\cr}
\$\$

```